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I am an high school student, and I'm diving deeper and deeper into Maths, and thinking into studying it at university. I have read multiple books, and gazed at the beautiful proofs presented there. In particular I enjoy to see proof by contradiction, such as the proof of the infinite number of prime numbers, and the elegant way they are presented.

However this made a question come to my mind: Is math fool-proof? In the specific I am wondering if it could exist a theorem whose only formulation would "break math", specifically whose trueness would bring a contradiction but whose falsehood would also create a contradiction.

Is such a thing possible?

WizardOfMenlo
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  • @NormalHuman which kind do you suggest? – WizardOfMenlo Oct 15 '15 at 13:51
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    It turns out, we know that we are unable to prove that math is consistent, in general. We can often prove if theory A is consistent then theory B is consistent, but it is theoretically possible for the whole structure to come crashing down on us. – Thomas Andrews Oct 15 '15 at 13:53
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    This should lead you down the rabbit hole nicely. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems – Jeffrey L. Oct 15 '15 at 13:54
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    "Is math fool-proof?" I know plenty of fools who do math. ;P –  Oct 15 '15 at 13:54
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    I like this question, and I think its better phrased than the question that I marked it as a possible duplicate of. Nonetheless, the answers there should answer your question. – goblin GONE Oct 15 '15 at 13:58
  • That was a lot of interest quite quickly :) – WizardOfMenlo Oct 15 '15 at 13:58

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A question that "broke math" in the past was whether there exists a set that consists of all sets that are not elements of themselves (aka, Russell's paradox). But the "break" was fixable. Though, as Thomas Andrews says, we cannot prove math is consistent (how could you, since such a proof would be part of math?), there is every reason to believe that any new true paradox would simply require an adjustment in the axioms to disallow it.

Paul Sinclair
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